Find a parametric representation for the surface: The plane through the origin that contains the vectors i - j and j - k. Hint: Let x, y, z be in terms of u and/or v.

Q: Sketch and describe, in your own words, the surface S parametrized as follows: r(u, v) = (3 cos v,…

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Q: Which of the following accurately described the parametrized surface below? O paraboloid O cone O…

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Q: Evaluate F.dr, where F (x, y, z) = – y?i + xj + z°k and C is the curve of intersection of the plane…

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Q: Let D be the solid in space lying beneath the ry-plane and between the sphere r2 + y? + z² = 3 and…

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Q: Match each parametrization with the corresponding surface. (i) (ii) (iii) (iv) (v) Answer Bank (u,…

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Q: Evaluate (x2 + y? + z)dS, where S is the portion of the plane z = x + 2 that lies inside the…

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Q: state the type of the quadric surface and describe the trace obtained by intersecting with the given…

A: x2+y42+ z2=1,y=5x2+y42+ z2=1

Q: Determine whether the statement is true or false. If D is the disk given by x2 + y2 s 49, then 49 x2…

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Q: Find a vector parametric equation for the ellipse that lies on the plane 5y−3x+z=9 and is inside the…

A: Given that: 5y-3x+z=9 Let x=u cos v, y=u sin v So z=9+3x-5y  =9+3u cos v-5u sin v

Q: Determine the constants a, B, y such that the surfaces ax?z – xy? = B, yaz – y² = 0, - intersect…

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Q: Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface.…

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Q: Find a vector function that represents the curve of the intersection of the two surfaces. The…

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Q: Find a parametric representation for the surface. The part of the sphere x2 + y2 + z2 = 16 that lies…

A: Given:   Equation of cone z=x2+y2   Equation of paraboloid z=x2+y2  Imagine a sphere of radius…

Q: Determine the parametric equations of the curve of intersection of the paraboloid x? – y? + z² = 0…

A: Detailed explanation mentioned below

Q: Sketch the quadric surface: r +4y2 +4z2 16. Note: The x, y, & z axes are depicted in red, green, &…

A: The given function is: x2+4y2+4z2=16

Q: Determine the intersection of the hyperbolic paraboloid z = (y2/b2) − (x2/a2) with the plane bx + ay…

A: Given The hyperbolic paraboloid equation is z=y2b2-x2a2 and the plane bx + ay − z = 0 .

Q: Find a vector that is perpendicular to the surface 4e^{(z-1)}+x^{2}y=xyz at the point (−1,−2,1).

A: Given the surface :

Q: the trace of the surface in the plane x=1?

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Q: Check whether the given surface (x – a)² + (y – b)² + (z – c)? 16 is developable or not. -

A: Given surface(x-a)2+(y-b)2+(z-c)2=16orz=c+16-(x-a)2-(y-b)2If the surface is developable…

Q: Show that the tangent plane to the quadric surface at the point (x0, y0, z0) can be written in the…

A: Given surface is x2a2+y2b2+z2c2=1 To find the equation of the plane, find the normal of the given…

Q: Evaluate ⌠⌠(x+y)ds, where S is the cone whose parametric equations are x=ucosv, y=usinv, z=2u,…

A: Given  cone whose parametric equations are x=ucosv, y=usinv, z=2u, 0≤u≤3, 0≤v≤π/2 we find the given…

Q: Find the point lying on the intersection of the plane x + y + 4 z = 0 and the sphere x2 + y2 + z2 =…

A: Determine z from the equation of plane.

Q: Let h, r be constants. Consider the quadric surface S of equation (x – h)? (y – 1)² (z – r)2 9 3 The…

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Q: Determine the projection of the surface - y = z on the xy-plane? (Hint: z = 0) O Hyperbola O Ellipse…

A: projection of the surface

Q: Write an equation for the plane tangent to the surface F(x, y, z) = 0 at the point (a, b, c).

A: Given surface is Fx, y, z=0 and the point is a, b, c.

Q: Does every second-degree equation in x, y, and z represent a quadric surface? Explain.

A: the equation of a quadric surface in space is a second degree equation in three variables. The…

Q: 1. Find a parametric representation for the surface: the part of the sphere x2 + y? + z2 = 4 that…

A: According to the given information, It is required to calculate the parametric representation for…

Q: ). Find a parametric representation for the line of intersection of the following two planes: I - y…

A: We use cross product to find the direct vector for the line.

Q: Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface…

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Q: Let o be the surface of the sphere given by r² + y? + z² = a². Evaluate the surface integral [S,(r²…

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Q: Let P(r, y, z) be the point at which the line with parametric equations x = -t, y=3+2t, z=1-t te R,…

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Q: Which of the following is a non-orientable surface? O a. all the other choices are orientable O b.…

A: orientable surface -  The surface for which we can define clockwise consistently.

Q: Find a parametric representation for the surface. the part of the plane  z = x + 2  that lies inside…

A: We need to find parametric representation.

Q: Describe and sketch the surfaces in space (R') defined by the following equa- tions: (a) y = -z +1…

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Q: Find the surface normal vector to the surface z 4-(x +y²) at the point (1, 1,v2) by (a) Vf (b) N =…

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Q: Sketch or describe the surfaces in R³ determined by the equa- tions in Exercises 40–46. z = y? + 2

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Q: Describe and sketch the surface in R3 represented by  a. 2x+z=3 b. z=cos y

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Q: Write an equation of the tangent plane to the parametric surface: r(u, v) = (u + v)i + 3u^2j + (u −…

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Q: Find the distance between two planes a and a1 that are tangent to the ellipsoid 2 +2y? + z² = 5/8…

A: We have to find the distance between the two tangent planes parallel to x+y+z=0  that are tangent to…

Q: If A = 18zî – 12ĵ + 3yk then find Jſ, A.ñ ds where S is the surface of the plane %3D 2x + 3y + 6z =…

A: Note "As per our company guidelines we are supposed to answer ?️only first question. Kindly repost…

Q: Identify the surface with the given vector equation. r(0,0) = 2sinocosei +2sinosinej +2/1+sin?ok a.…

A: The minimum number of parameters needed to represent a surface is two. Therefore, in general, the…

Q: Determine whether the statement is true or false. If D is the disk given by x2 + y2 < 4, then I V4-…

A: Given the given relation is

Q: Which of the following represents a plane surface whose x-, y-, and z-intercepts are a, b, and c…

A: Given questions are solved.

Q: Express the area of the given surface as an iterated double inegral in polar coordiantes, and then…

A: We need to find the surface area in the portion of the sphere x^2+y^2+z^2=16 between the planes z=1…

Q: Find an equation of the plane that is normal to the surface Iz + yz? = 2+ yz at the point (2,-1, 1).

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Q: Find a parametric representation for the surface. the part of the sphere x2 + y? + z? = 16 that lies…

A: The solution is below.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.